Experimental identification of nonlinear dynamic properties of built-up structures L. Heller a, , E. Folte ˆte b , J. Piranda b a Institute of Physics ASCR, v.v.i., Na Slovance 2, Prague 18221, Czech Republic b Institute FEMTO-ST, Laboratory of Applied Mechanics, University of Franche-Compte´, 25000 Besanc - on, France article info Article history: Received 26 October 2007 Received in revised form 30 May 2009 Accepted 2 June 2009 Handling Editor: C.L. Morfey Available online 28 June 2009 abstract The paper is focused on the nonlinear damping capacity of built-up structures owing to presence of frictional joints. To quantify and characterise the nonlinear dynamic behaviour of built-up structures, a novel experimental procedure is introduced based on the wavelet transform providing equivalent modal parameters identification. In order to analyse the influence of the interfacial pressure and the interface area on the dynamic behaviour of built-up structures, an experimental study conducted on a simple built-up structure consisting of two bolted beams is presented. Results in terms of equivalent modal parameters are finally discussed and rationalised. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Complex mechanical systems are generally composed of simple elements assembled by connections such as bolted, riveted and welded joints. The presence of those mechanical joints affects the dynamic behaviour in terms of eigenfrequency and damping [1,2]. Moreover, the complex transfer behaviour of mechanical joints introduces nonlinearities into the dynamic response of assembled structures. This influence has to be taken into account during the stage of engineering design in order to predict and then optimise the dynamic behaviour. Mechanical joints have also been recognised as one of the main sources of the energy dissipation in complex built-up structures [3,4]. Although this sort of dissipation can be related to many physical phenomena [1], friction between the substructures is considered the most important [2]. The term slip damping is used to refer to this mechanism. The earliest studies of the slip damping were devoted to analysis of simple built-up structures. The joints were idealised by introducing assumptions such as a uniform interfacial pressure [5–7]. Experimental analyses were accompanied by analytical calculations leading to mathematical relations linking the energy dissipation or slip damping to different parameters such as friction coefficient, contact pressure, amplitude of loading, etc. Interesting results coming out of these studies were the existence of an optimal pressure value giving the maximal slip damping [5] and the slip damping dependence proportional to the cube of the loading force amplitude [4]. One of the most important earlier experimental works on slip damping was carried out by Ungar [1]. In his work light assembled structures of the type used in aircrafts were analysed in order to find the origin of the energy dissipation taking place in mechanical joints. The influence of the type of structure and frequency on slip damping was also studied. Ungar proposed a number of semi-empirical equations providing the evaluation of the loss energy factor for some specific structures. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration ARTICLE IN PRESS 0022-460X/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2009.06.008 Corresponding author. E-mail address: [email protected] (L. Heller). Journal of Sound and Vibration 327 (2009) 183–196
Transcript
ARTICLE IN PRESS
Contents lists available at ScienceDirect
Journal of Sound and Vibration
Journal of Sound and Vibration 327 (2009) 183–196
0022-46
doi:10.1
� Cor
E-m
journal homepage: www.elsevier.com/locate/jsvi
Experimental identification of nonlinear dynamic properties ofbuilt-up structures
L. Heller a,�, E. Foltete b, J. Piranda b
a Institute of Physics ASCR, v.v.i., Na Slovance 2, Prague 18221, Czech Republicb Institute FEMTO-ST, Laboratory of Applied Mechanics, University of Franche-Compte, 25000 Besanc-on, France
a r t i c l e i n f o
Article history:
Received 26 October 2007
Received in revised form
30 May 2009
Accepted 2 June 2009
Handling Editor: C.L. MorfeyAvailable online 28 June 2009
The paper is focused on the nonlinear damping capacity of built-up structures owing to
presence of frictional joints. To quantify and characterise the nonlinear dynamic
behaviour of built-up structures, a novel experimental procedure is introduced based on
the wavelet transform providing equivalent modal parameters identification. In order to
analyse the influence of the interfacial pressure and the interface area on the dynamic
behaviour of built-up structures, an experimental study conducted on a simple built-up
structure consisting of two bolted beams is presented. Results in terms of equivalent
modal parameters are finally discussed and rationalised.
& 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Complex mechanical systems are generally composed of simple elements assembled by connections such as bolted,riveted and welded joints. The presence of those mechanical joints affects the dynamic behaviour in terms ofeigenfrequency and damping [1,2]. Moreover, the complex transfer behaviour of mechanical joints introducesnonlinearities into the dynamic response of assembled structures. This influence has to be taken into account duringthe stage of engineering design in order to predict and then optimise the dynamic behaviour. Mechanical joints have alsobeen recognised as one of the main sources of the energy dissipation in complex built-up structures [3,4]. Although thissort of dissipation can be related to many physical phenomena [1], friction between the substructures is considered themost important [2]. The term slip damping is used to refer to this mechanism.
The earliest studies of the slip damping were devoted to analysis of simple built-up structures. The joints were idealisedby introducing assumptions such as a uniform interfacial pressure [5–7]. Experimental analyses were accompanied byanalytical calculations leading to mathematical relations linking the energy dissipation or slip damping to differentparameters such as friction coefficient, contact pressure, amplitude of loading, etc. Interesting results coming out of thesestudies were the existence of an optimal pressure value giving the maximal slip damping [5] and the slip dampingdependence proportional to the cube of the loading force amplitude [4].
One of the most important earlier experimental works on slip damping was carried out by Ungar [1]. In his work lightassembled structures of the type used in aircrafts were analysed in order to find the origin of the energy dissipation takingplace in mechanical joints. The influence of the type of structure and frequency on slip damping was also studied. Ungarproposed a number of semi-empirical equations providing the evaluation of the loss energy factor for some specificstructures.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196184
More recently, analysis of the slip damping has been oriented to more complex built-up structures. The use of numericalmethods such as the finite element method allows to perform complex nonlinear contact analysis giving an insight into thedistribution and amount of friction taking place in joints during vibrations. However, these analyses are very difficult tocarry out owing to particularly large models that result from a fine discretisation of contact surfaces guaranteeing thenumerical stability. Hence, several approaches have been suggested to simplify the numerical analysis of assembledstructures. One of the approaches is to apply constitutive models of mechanical joints that use degrees of freedom naturalto the scale of structural dynamics. Many constitutive models have been proposed such as Valanis model [8], Iwan model[9,10], Ren’s model [11] and the point contact model worked out by Sanliturk and Ewins [12]. The constitutive models arefully defined by a certain number of parameters depending on properties of a given joint. Their determination can be doneby using either experimental [9] or numerical approaches [10].
Another way to represent the nonlinear transfer behaviour of mechanical joints is by extending the modal analysis tononlinear structures. Ferreira and Ewins proposed a method of nonlinear impedance based on the Multi-HarmonicDescribing Function [13]. The method consists of taking into account the harmonic oscillations at multiples of theexcitation frequency.
The frequency response function (FRF) depending on the excitation amplitude was proposed by Siller [14] whodeveloped a method for detecting, localising, identifying and quantifying the nonlinearities using the amplitude dependentFRF.
Finally, equivalent modal parameters [15] representing the variation of modal parameters with respect to the vibrationamplitude has been used [16,17] in order to characterise the dynamical nonlinearities introduced into the system throughmechanical joints.
In this paper, we use the equivalent modal parameters to assess the influence of mechanical joints on the dynamicbehaviour of assembled structures. First, a mathematical definition of the equivalent modal parameters is introduced forthe case of a general mdof system having non-coupled eigenmodes. Then an experimental method for identification of theequivalent modal parameters is proposed. The identification is based on analysis of free-decay responses using the wavelettransform.
The last part of the paper describes an experimental study dealing with a simple built-up structure represented by twooverlapped bolted beams. The proposed identification method is applied to analyse the dependence of the equivalentmodal parameters on the vibration amplitude, interfacial pressure and surface of the contact area.
2. Definition of equivalent modal parameters
In this paper, we focus on assembled structures consisting of linear substructures. Thus those structures can beconsidered as dynamic systems locally nonlinear due to the friction in joints. Furthermore, we assume that the friction doesnot substantially affect the eigenvector linearity. To characterise dynamic systems that meets all these assumptions, theapproach of equivalent modal parameters was chosen. According to the well known small-parameter method proposed byKrylov and Bogoljubov [15], the eigenfrequency and modal damping of a locally nonlinear system can be seen asparameters depending on the vibration amplitude. Evolutions of the equivalent modal parameters with respect to theamplitude of vibrations provide information about both the type and the degree of nonlinearities of analysed systems.
The dynamic free motion of a locally nonlinear structure can be approximated by a discrete model whose free dynamicresponse is defined as a superposition of free motions of N single dof nonlinear oscillators as follows:
x ¼ Vq, (1)
xi ¼XNj¼1
Vi;jajðtÞ cosYjðtÞ ¼Xj¼1
AijðtÞ cosYjðtÞ, (2)
where x, V and q denote the displacement vector, the modal matrix containing all eigenvectors and the vector of modalcoordinates, respectively. A particular modal coordinate qj is characterised by its amplitude aj and phase Yj.
According to the small-parameter method, the equivalent modal eigenfrequency and damping of N single dof nonlinearmodal oscillators are defined by the following set of differential equations:
where the damped eigenfrequency d;eOj and the modal damping ezj represent the set of equivalent modal parameters and �denotes the small parameter. As in the case of linear dynamic systems, the equivalent modal damped eigenfrequency islinked to the non-damped equivalent modal eigenfrequency n;eOi and equivalent damping coefficient ezi.
It should be pointed out that the present definition of equivalent modal parameters (Eqs. (3) and (4)) is valid only forsystems having non-coupled eigenmodes. In such a case, the contribution of adjacent eigenmodes can be neglected when
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196 185
systems vibrate at frequencies close to a given eigenmode. Under those assumptions, an experimental method for theequivalent modal parameters identification was developed.
3. Experimental identification of equivalent modal parameters
The proposed experimental method extracts the equivalent modal parameters from the free-decay system responsewhich is assumed to satisfy Eq. (2).
The identification is based on the use of the continues wavelet transform [18,19] which is applied on the free-decay timeresponse. Many experimental studies utilising a wavelet based identification for linear [20–23] and nonlinear systems[24,25] have been published in past two decades. Therefore, only a basic concept of our identification method will bedescribed.
The projection of a free-decay time response provided by the ith accelerometer to the time–frequency domain is definedby the following wavelet transform formula:
Txiðs; tÞ ¼ hxiðtÞ;cs;ti ¼
1ffiffisp
Z 1�1
xiðtÞct � t
s
� �dt, (5)
where s, t, cððt � tÞ=sÞ are the scaling parameter being reciprocally related to the frequency, the time shift and the complexconjugate shifted and scaled mother wavelet function, respectively. The mother wavelet represents a specific functionsatisfying many mathematical conditions [18,19] such as its well localisation in time and frequency. The choice of aparticular mother wavelet is determined by the nature of the time signal to be treated. In our study we deal with oscillatingsignals being well localised in frequency which implies that the mother wavelet has to be an oscillating function being welllocalised in frequency. One of the wavelet functions satisfying this requirement is the Morlet function chosen as the motherwavelet for our identification method. The analytical solution of the wavelet transform formula Eq. (5) can be found underthe assumption of the asymptotic property of the analysed free-decay response xiðtÞ. The physical meaning of such aproperty is that the rate of phase change is much more important than that of amplitude change. If this assumption issatisfied, the free-decay response is projected to the time–frequency domain in the form of a complex 2D functiondescribed approximately by the following equation:
Txiðs; tÞ �
XNj¼1
ffiffisp
2Vi;jajðtÞcðsoðtÞÞe
iðYjðtÞÞ, (6)
where cðsoðtÞÞ is the complex conjugate Fourier transform of the wavelet function. The absolute value of complex wavelettransform coefficients can be interpreted as a surface in the time–frequency space as shown in Fig. 1.
The main feature of such a surface is the presence of ridges corresponding to the system eigenfrequencies (Fig. 1). Aridge characterises the variation of a particular eigenfrequency in time that can be considered as an indicator of thestiffness linearity. Such a linearity may be approved by existence of a unique scale value determining the position of theridge within the entire time interval.
Many algorithms and methods providing the ridge identification and extraction have been proposed [18,19]. Thesimplest one based on the maximum absolute value of the wavelet transform coefficients is used in our study. Consider aridge sa;jðtÞ which corresponds to the jth eigenfrequency. In practical computations, the wavelet transform is calculatedonly for a discrete vector of scales s1 . . . sn. Then the jth ridge laying within a scale interval sa . . . sb is considered tocorrespond to scales satisfying the following equation:
Txiðsa;jðtÞ; tÞ ¼ maxðjTxi
ðsa:::sb; tÞjÞ. (7)
Fig. 1. Example of the time–frequency representation of a free-decay response: (a) time response, (b) module of wavelet transform coefficients.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196186
The jth component of the signal amplitude AijðtÞ and phase YjðtÞ identified at the ith accelerometer can be reconstructed
using the wavelet coefficients corresponding to the ridge scales by applying the following equations:
AijðtÞ ¼
2ffiffiffiffiffiffiffiffiffiffiffiffisa;jðtÞ
q jTxiðsa;jðtÞ; tÞj, (8)
YjðtÞ ¼2ffiffiffiffiffiffiffiffiffiffiffiffi
sa;jðtÞq argðTxi
ðsa;jðtÞ; tÞÞ. (9)
The introduction of Eqs. (8) and (9) into Eqs. (3) and (4) leads to Eqs. (10) and (11) showing that the equivalent modaldamping is linked to the logarithm of the module of wavelet coefficients and the equivalent damped eigenfrequency isrelated to the argument of these coefficients:
ezjiðtÞ ¼ �
1
n;linOj
d ln2ffiffiffiffiffiffiffiffiffiffiffiffi
sa;jðtÞq jTx;iðsa;jðtÞ; tÞj
0B@
1CA
dt; j ¼ 1;2; . . . ;N, (10)
d;eOjiðtÞ ¼
d argðTxiðsa;jðtÞ; tÞÞ
dt; j ¼ 1;2; . . . ;N. (11)
Note that the computation of Eqs. (10) and (11) requires the use of a numerical derivation scheme being a very sensitiveoperation when it is applied on standard measured signals containing a certain level of noise. Therefore, the derivative wascomputed using the Savitzky–Golay FIR (Finite Impulse Response) smoothing filter.
The time evolution of equivalent modal parameters can be converted to corresponding amplitude dependencies byusing the time variation of amplitude Ai
jðtÞ evaluated by Eq. (8). When several free-decay responses xi; i ¼ 1; . . . ; k, fromdifferent places of the structure are provided, the proposed experimental method allows to identify non-normalisedeigenvectors. They are related to both the ratio of the absolute value of the wavelet ridge coefficients and their relativephase shifts. Consider two free-decay responses xi, xk and the ridge scales sa;j corresponding to the jth eigenfrequency. Thenthe kth component of the jth non-normalised eigenvector is calculated by the following formulas:
jak;jðtÞj ¼Txkðsaj
; tÞ
Txiðsaj
; tÞ
����������, (12)
argðTxkðsaj
; tÞÞ � argðTxiðsaj
; tÞÞ ¼ 0! ak;jðtÞ ¼ jak;jðtÞj, (13)
argðTxkðsaj
; tÞÞ � argðTxiðsaj
; tÞÞ ¼ �p! ak;jðtÞ ¼ �jak;jðtÞj. (14)
After having normalised those eigenvectors they can be assembled into the modal matrix Vi;jðtÞ which may be used tocalculate the eigenmode amplitude ajðtÞ as follows:
ajðtÞ ¼ AijðtÞ=Vi;jðtÞ. (15)
Finally, all the equivalent modal parameters identified on each accelerometer can be evaluated as functions of theeigenmode amplitude (Eqs. (16) and (18)):
ezjiðtÞ�!ezj
iðajðtÞÞ, (16)
d;eOjiðtÞ�!d;eOj
iðajðtÞÞ, (17)
Vi;jðtÞ�!Vi;jðajðtÞÞ. (18)
The identified equivalent modal eigenmodes allow for a verification of the assumption of quasi-constant eigenmodes thatrepresents the basic hypothesis the proposed identification method relies on. Furthermore, the identification can beapproved by comparing the equivalent modal parameters identified on different accelerometers that should betheoretically identical. Thus those experimentally identified parameters are required to be equal within admissibleidentification error tolerances Dz, DO as expressed by
ezjiðajðtÞÞ ¼ ezj
kðajðtÞÞ �Dz ¼ ezjðajðtÞÞ � Dz 8i; k, (19)
d;eOjiðajðtÞÞ ¼ d;eOj
kðajðtÞÞ � DO ¼ d;eOjðajðtÞÞ � DO 8i; k. (20)
ARTICLE IN PRESS
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196 187
4. Experimental investigation of a simple built-up structure
A simple built-up structure with an isolated frictional joint was analysed by using the previously presentedexperimental identification method. As it has been mentioned, the interfacial pressure and the interface area are the crucialfactors determining the final impact of the frictional joint on the dynamic behaviour of the whole structure. Therefore, theexperiment was conducted in a way allowing to control those two parameters in order to study their influence on theevolution of equivalent modal parameters.
4.1. Experimental set-up
The analysed structure consisted of two steel beams of dimensions 700� 50� 15 mm and 700� 50� 5 mm. The beamswere assembled by means of three bolts M6 distributed along the beams length. A tube equipped with a strain gauge wasinserted between the head of the bolt and the beam (Fig. 2) in order to provide the measurement of the axial force in bolts.Although the axial force gives only an indirect information about the interfacial pressure, the term prestress will be used inthe following text to refer to axial force values measured by the strain gauges.
Four different beam assemblages were tested to evaluate the effect of the interface area:
�
direct assemblage of two beams (assemblage w0); � washers inserted between the beams at bolt axes;� small washer with an outer diameter of 8 mm (assemblage w1);� intermediate washer with an outer diameter of 12 mm (assemblage w2);� large washer with an outer diameter of 16 mm (assemblage w3).
In order to simulate free-free boundary conditions, the structure was suspended by means of two wires connected to one ofthe beams at approximately calculated node positions of the first bending eigenmode. The experimental configurationincluding measurement equipment is shown in Fig. 2. Although the identification method was based on analysis of free-decay dynamic responses, an electromagnetic shaker was used for the dynamic excitation located at the extremity of thebeams where a force transducer and a piezoelectric accelerometer were also placed. Four other accelerometers weredistributed along the beam length allowing the eigenmode identification. The use of the electromagnetic shaker forexcitation purposes instead of e.g. hammer allowed to enlarge the amplitude interval within which the equivalent modalparameters were identified by using the presented identification method. It was achieved by using a sinusoidal excitationat a frequency close to the desired one that emphasised the presence of the particular eigenfrequency in the responsespectrum. The free-decay system response was then obtained by using a system disconnecting the shaker from thestructure. This system was realised by means of an electromagnet attached to the bar of the shaker and connecting theforce transducer and accelerometer that were both fixed on the structure. An electromagnet power supply control systemlinked to the data acquisition system was used for synchronising the time of disconnection with the start of the dataacquisition.
The complete measurement set-up scheme showing all acquired data is depicted in Fig. 3. Note that an indirectmeasurement of the interfacial pressure variation during the vibration was allowed by acquiring the strain gauge signalsduring vibrations.
Fig. 2. Experimental set-up and design of two assembled beams.
ARTICLE IN PRESS
Fig. 3. Experimental set-up scheme with all measurement equipments.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196188
4.2. Measurement technique
The experiment on the assembled beams was focused on the identification of equivalent modal parameterscorresponding to the first bending mode. The measurement procedure used for such an identification consisted of twomain steps. First, the structure was excited by a harmonic signal at a frequency close to that of the first eigenmode and thenthe free-decay response was recorded after the shaker disconnection from the structure by using the system describedabove. The second step consisted of applying the proposed wavelet based equivalent modal parameters identification onthe recorded free-decay responses. The overall identification technique comprised the following identification steps:
�
computation of the global wavelet transform using a large frequency band; � localisation of the ridge corresponding to the first eigenfrequency; � computation of the local wavelet transform using a narrow frequency band around the localised eigenfrequency; � identification of the first eigenfrequency ridge; � computation of the time evolution of equivalent modal parameters by using the wavelet transform coefficients
corresponding to the first eigenfrequency ridge;
� filtering of the free-decay response for obtaining the first eigenmode component, this step was performed by a pseudo-
inverse wavelet transform which is the standard inverse wavelet transform that uses only those wavelet coefficientsthat correspond to the first eigenfrequency ridge;
� conversion of the filtered acceleration response to displacement, this step allowed the conversion from the time
dependencies of equivalent modal parameters to their variations with respect to the displacement amplitude.
4.3. Preliminary modal analysis
First, a preliminary modal analysis using a random noise excitation technique was carried out in order to identify thefirst bending mode and analyse the influence of prestress on the FRF. Fig. 4 shows a shift of the first eigenfrequency towardshigher values when increasing the prestress. Such an effect indicates an increasing stiffness of the assembled structure.Furthermore, the half-power bandwidth decreases when increasing the prestress. Moreover, the FRFs of differentassemblages measured at a same applied axial force of 500 N (Fig. 5) show considerable differences in damping expressedby a varying half-power bandwidth. The beams in direct contact display a much higher value of damping than thatidentified in the case of assemblages with inserted washers. Although the FRFs of beams with inserted washers of differentsizes reveal only small changes in the half-power bandwidth, the damping clearly increases with increasing washer size.Hence, as the prestress and the interface area influence the amount of the relative motion between two beams, theexperimental results suggest that changes in the friction taking place at the frictional joint are the main cause of observedvariations in the equivalent modal parameters.
4.4. Eigenmode nonlinearity
In order to evaluate the impact of the interface area on the eigenvector linearity, a linear finite element model of theassemblage without washers was prepared (Fig. 6) so that the coincident interfacial nodes of two beams were fixed one to
ARTICLE IN PRESS
Fig. 4. FRFs measured on beams in direct contact tightened by different bolt axial forces.
Fig. 5. FRFs obtained on all tested assemblages tightened by an axial force of 500 N.
Fig. 6. The finite element mesh of a linearised FEM model and the calculated first eigenmode.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196 189
the other i.e. the friction and contact were not considered in the computation. The first bending mode shown in Fig. 6 wasthen compared to that identified experimentally with respect to the vibration amplitude as it has been explained above.
Figs. 7 and 8 show a comparison between the calculated eigenmode and the mean value of the measured one bothexpressed in the normalised form. The comparison showing a relative error never exceeding 16 percent proves that thefrictional interface of the bolted joint has only a slight influence on the shape of the first eigenvector. Nevertheless, theerror distribution along the beam length is not symmetric which might suggest a slight deviation from the eigenmodesymmetry due to the frictional joint. Then the normalised experimentally identified eigenvector Vi;1ðtÞ; i ¼ 1; . . . ;5, wasused to convert the identified vibration amplitudes Ai
1; i ¼ 1; . . . ;5, from all accelerometers to the eigenmode amplitudea1ðtÞ ð� aðtÞÞ as discussed in Section 3 so that we could express all identified modal parameters with respect to theeigenmode amplitude.
After having identified the eigenmode amplitude, the experimental identification of equivalent modal parameters wasapproved by comparing data from measurements on different accelerometers that were supposed to give the same resultswithin an identification error tolerance as expressed by Eqs. (19) and (20). The identified results satisfied this condition inall measured configurations, an example of this agreement is depicted in Fig. 9 showing the results for an axial force of5000 N and an assemblage of type w0.
As explained previously, the proposed experimental identification of equivalent modal parameters is developed underthe assumption of an eigenmode quasilinearity. Thus to verify such a linearity, the amplitude dependence of the relativedeviation of eigenvector components DrVi;1ðaðtÞÞ from their values at the minimal vibration amplitude was evaluated usingthe following equation:
DrVi;1ðaðtÞÞ ¼Vi;1ðaðtÞÞ � Vi;1ðaminÞ
jVi;1ðaminÞj, (21)
ARTICLE IN PRESS
Fig. 7. Comparison of the first eigenmode obtained by FE analysis and measurement.
2 3 4 50
2
4
6
8
10
12
14
16
accelerometer
rela
tive
erro
r [%
]
1000N5000N10000N
Fig. 8. Relative error between the calculated and the averaged experimentally obtained the first eigenmode.
Fig. 9. Comparison of equivalent modal parameters identified on different accelerometers.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196190
where Vi;1ðaminÞ denotes the value of the ith component of the first eigenvector at the minimal vibration amplitude reachedwithin free vibrations, whereas Vi;1ðaðtÞÞ represents its instantaneous value.
The evolution of the relative deviation of the third component of the first eigenvector component is shown in Fig. 10 as arepresentative example. The recorded data were affected by the presence of signal noise (see markers in Fig. 10representing data points) and therefore a curve fitting was applied on the measured values (solid lines in Fig. 10) tohighlight the trend of plotted evolutions. In the case of a linear system no amplitude dependence of eigenvectors existswhich would correspond to dashed line in Fig. 10. In contrast, the present dynamic system clearly shows an amplitude
ARTICLE IN PRESS
Fig. 10. Amplitude dependence of the relative deviation of the third component of the first eigenvector from its value reached at the minimal vibration
amplitude. The dashed line indicates the evolution in the case of an ideal linear system.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196 191
dependence of the measured eigenvector. This dependence becomes more important with decreasing prestress that mightbe interpreted as increasing eigenmode nonlinearity when the role of friction in the dynamic behaviour of the systembecomes more important. In the case of higher prestresses (10 000 and 5000 N) only a slight maximum absolute value ofthe relative deviation of the eigenvector component was identified ðo1Þ percent that was reached already at smallamplitudes and did not increase further. However, in the case of a very low prestress (1000 N) the effect of friction on theeigenvector amplitude dependence seems to be more important as the absolute value of the relative deviation shows ahigher value ð3:5Þ percent and an increasing tendency with increasing amplitude. Nevertheless, as the identified relativedeviation of the eigenvector components is limited to few percents, the assumption of the eigenmode quasilinearity seemsto be satisfied in the case of the analysed dynamic system.
The eigenvectors could also be affected by a modal coupling due to the nonlinear damping forces. Nevertheless, the verysmall relative errors indicate that this coupling did not occur, although the orthogonality of the eigenmodes was notverified.
A negligible eigenvector variation with respect to the amplitude and prestress justifies the use of the proposedidentification method relying on the Single Degree of Freedom approach. In that situation, a single sensor located anywhereon the structure except on a nodal line allows the identification of the modal equivalent parameters. It is clear that thisapproach would no longer be valid in the case of a modal coupling due to the nonlinear damping forces. In that case, aMulti-Degree of Freedom technique would be necessary and the questions of the number and locations of sensors wouldhave to be addressed.
4.5. Evolution of equivalent eigenfrequency
As each of studied assemblages has a slightly different geometry, the identified damped eigenfrequencies do not lie atthe same frequency level (Fig. 11). However, all assemblages exhibit a nearly linear decrease of the eigenfrequency whenincreasing the vibration amplitude. The slope of such a decrease depends on the assemblage type i.e. on the interface area.This would suggest that the amplitude dependence of the eigenfrequency can be attributed to the friction at the interface.
The nonlinearity degree of the damped eigenfrequency expressed by the slope of its amplitude dependence dramaticallychanges when decreasing the axial force as can be seen in Fig. 12. It may be explained by the fact that slightly tightenedbolts allow for an important friction motion giving rise to the slip damping that increases with increasing amplitude ofvibrations. Consequently, such a vibration amplitude sensitivity of the slip damping is directly projected to the amplitudeevolution of the damped eigenfrequency through their relation given by Eq. (4). As the applied prestress increases, thedamped eigenfrequency becomes almost constant due to a restricted friction at the interface that limits the role of the slipdamping. This stabilisation or linearisation of the eigenfrequency is better represented by the prestress dependence of theeigenfrequency for different amplitude levels as depicted in Fig. 13 which shows data corresponding to the assemblage w0(no washer inserted). The damped eigenfrequency asymptotically tends to a same limit value for all amplitudes ofvibration. Such a limit value can be considered as the eigenfrequency of a corresponding virtual linear system being free offriction.
4.6. Evolution of equivalent damping
The identified equivalent damping showed in Fig. 14 reveals a clear relation between the interface area and the dampinglevel. It increases with increasing interface area for all applied prestresses at a given vibration amplitude. The equivalentdamping of the assemblage w0 having beams in direct contact was found to be several times higher than in the case of
Fig. 13. Asymptotic evolution of the damped eigenfrequency with increasing prestress.
axial force - 500 N
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.5
1
1.5
2
2.5
a [mm]
ζ e [%
]
axial force - 2000 N
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
0.5
1
1.5
a [mm]
ζ e [%
]
axial force - 6000 N
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a [mm]
ζ e [%
]
axial force - 10000 N
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a [mm]
ζ e [%
]w1w2w3w0
w1w2w3w0
w1w2w3w0
w1w2w3w0
Fig. 14. Effect of the interface area on the vibration amplitude evolution of the equivalent damping coefficient at different applied prestresses.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196 193
assemblages with inserted washers. It would suggest that the slip damping plays an important role in the overall energydissipation of the structure and that the area where an important friction occurs is not limited to a small perimeter aroundthe bolt but is larger than the diameter of the largest washer ðd ¼ 16 mmÞ.
Fig. 15 also shows an important effect the prestress has on the character of the equivalent damping variation withrespect to the vibration amplitude. The equivalent damping variation corresponding to lower prestresses seems to satisfy apower law while a linear evolution is observed in the case of higher prestresses. Moreover, at low prestresses, theassemblages with inserted washers show damping variations satisfying a power law and a local maximum at higheramplitude seems to exist. It could be explained by the existence of a limited perimeter around the bolt axis beyond whichthe interfacial pressure vanishes. Hence, when the friction area reaches this limit upon increasing vibration amplitude, anyfurther amplitude increase is not accompanied by a damping increase. This local maxima in the equivalent damping
ARTICLE IN PRESS
0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
0.5
1
1.5
2
2.5assemblage w0
a [mm]
ζ e [%
]assemblage w1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.2
0.4
0.6
0.8
1
1.2
a [mm]
ζ e [%
]
assemblage w2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.20.40.60.8
11.21.41.61.8
a [mm]
ζ e [%
]
assemblage w3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
0.20.40.60.8
11.21.41.6
a [mm]
ζ e [%
]
10000 N8000 N6000 N4000 N2000 N500 N
10000 N8000 N6000 N4000 N2000 N500 N
10000 N8000 N6000 N4000 N2000 N500 N
10000 N8000 N6000 N4000 N2000 N500 N
Fig. 15. Effect of the axial bolt force on the vibration amplitude evolution of the equivalent damping coefficient corresponding to different assemblage
types.
0 1500 3000 4500 6000 7500 9000 100000.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
axial force [N]
ζ e [%
]
vibration amplitude
Fig. 16. Asymptotic evolution of the equivalent damping coefficient with increasing prestress.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196194
evolution should also exists in the case of the assemblage with direct contact owing to a presumable local distribution ofthe interfacial pressure around the bolt axis. Such a localised pressure should also posses a limited perimeter beyond whichthe pressure vanishes. Therefore, when the continuing increase of vibration amplitude would deploy the friction areabeyond this perimeter, the dissipation by friction would no longer increase. In order to verify this hypothesis, anexperiment with a sufficiently strong excitation deploying the friction area over such a perimeter would have to be done.Unfortunately, the used electromagnetic shaker was not sufficiently powerful to allow such an excitation.
Besides qualitative changes in damping variations with respect to the prestress, one can also observe importantquantitative damping changes when decreasing the prestress. In the case of assemblage w0, an increase of damping as
ARTICLE IN PRESS
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196 195
large as 500 percent was observed upon a prestress decrease from 10 000 to 500 N. As the friction between two beams isstrongly conditioned by the applied prestress, those results proves that the frictional joints may provide a high passivedamping capacity when the relative motion between substructures is not fully restricted.
Finally, an asymptotic damping stabilisation due to increasing prestress was identified as in the case of the dampedeigenfrequency. Fig. 16 shows that when the prestress grows up the equivalent damping asymptotically tends to a samevalue for any applied vibration amplitude. In other words, the damping is getting linear upon increasing prestress owing toa restricted nonlinear slip damping.
5. Conclusions
The aim of this paper was to analyse the influence of frictional joints and their functional parameters on the globaldynamic behaviour of built-up structures. For such a purpose an experimental study was carried out on a simple built-upstructure consisting of two bolted beams designed so that the prestress and the interface area could be modified. Tocharacterise its presumably nonlinear dynamic behaviour, the equivalent modal parameters were chosen. Theidentification of equivalent modal parameters was performed by a new experimental method developed under theassumptions of linear substructures and non-coupled quasilinear eigenmodes. The method is based on the use ofthe wavelet transform applied on the free-decay response. This method was used in the experiment for analysing the globaldynamic behaviour of the studied assembled structure with regard to the functional parameters of the frictional jointrepresented by the prestress and the interface area. The results and main conclusions of the experiment may besummarised as follows:
�
the frictional joint affects the dynamic behaviour of the whole structure only within a limited range of bolt axial forcesand vibration amplitudes; � the frictional joint influences only negligibly the eigenmode linearity at all applied prestresses and vibration
amplitudes;
� at low bolt axial forces (up to 2000 N), the frictional joint gives rise to an important slip damping resulting in nonlinear
amplitude dependent equivalent modal parameters;
� the equivalent modal parameters tend to stabilise at constant amplitude independent values when increasing the bolt
axial force beyond a limit value;
� the equivalent eigenfrequency shows a linearly decreasing tendency with increasing amplitude at low bolt axial forces; � at low bolt axial forces, the equivalent damping is strongly dependent on the vibration amplitude, bolt axial forces and
interface area;
� at important vibration amplitudes, the equivalent damping reaches extreme values that are several times higher
compared to its stabilised amplitude independent value at high axial bolt forces;
� the extreme values of the equivalent damping increases substantially with increasing interface area.
The experiment confirmed that the frictional joints are main sources of energy dissipation in built-up structures whenapplied prestresses allow for a relative motion of substructures. Under those circumstances the studied structure showed ahigh passive damping capacity as large as 2.5 percent that was five times more than under highly tightened conditions. Theexperiment also proved that the friction is not restricted at a localised area around the bolt but it is extended to muchlarger interface area. Hence, the passive damping capacity may be considerably increased by extending the interface area.
References
[1] E.E. Ungar, Energy dissipation at structural joints; mechanisms and magnitude, Air Force Flight Dynamics Laboratory, Wright-Pattersons Air ForceBase, OH, USA, 1964.
[2] L. Gaul, R. Nitsche, The role of friction in mechanical joints, Applied Mechanics Reviews 54 (2) (2001) 93–106.[3] E.E. Ungar, The status of engineering knowledge concerning the damping of built-up structures, Journal of Sound and Vibration 26 (1) (1973) 141–154.[4] L.E. Goodman, A review of progress in analysis of interfacial slip damping, Structural Damping, American Society of Mechanical Engineers, 1959,
pp. 35–48.[5] L.E. Goodman, J.H. Klumpp, Analysis of Slip Damping with Reference to Turbine-blade Vibration, ASME Applied Mechanics Division, vol. 23, 1956,
pp. 421–429.[6] T.H.H. Pian, Structural Damping of a Simple Built-up Beam with Riveted Joints in Bending, ASME Applied Mechanics Division, Vol. 24, 1957, pp. 35–38.[7] M. Masuko, Y. Ito, K. Yoshida, Theoretical analysis for damping ratio of a jointed cantibeam, Bulletin of the JSME 16 (1973) 1421–1432.[8] L. Gaul, J. Lenz, Nonlinear dynamics of structures assembled by bolted joints, Acta Mechanica 125 (1–4) (1997) 169–181.[9] Y. Song, C.J. Hartwigsen, D.M. McFarland, A.F. Vakakis, Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam
elements, Journal of Sound and Vibration 273 (2004) 249–276.[10] D.J. Segalman, An initial overview of Iwan modelling for mechanical joints, Sandia Report SAND2001-0811, Sandia National Laboratories, New Mexico
and Livermore, CA, USA, 2001.[11] Y. Ren, C.F. Beards, Identification of joint properties of a structure using FRF data, Journal of Sound and Vibration 186 (4) (1995) 567–587.[12] K.Y. Sanliturk, D.J. Ewins, Modelling two-dimensional friction contact and its application using harmonic balance method, Journal of Sound and
Vibration 193 (1996) 511–523.[13] J.V. Ferreira, Dynamic Response Analysis of Structures with Nonlinear Components, Department of Mechanical Engineering, Imperial College of
Science, Technology and Medicine, London, UK, 1998.
L. Heller et al. / Journal of Sound and Vibration 327 (2009) 183–196196
[14] H.R.E. Siller, Nonlinear Modal Analysis, Methods for Engineering Structures, Department of Mechanical Engineering, Imperial College of Science,Technology and Medicine, London, UK, 2004.
[15] A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.[16] M. Feldman, Non-linear system vibration analysis using Hilbert transform—part 1: free vibration analysis method FREEVIB, Mechanical Systems and
Signal Processing 8 (2) (1994) 119–127.[17] C.J. Hartwigsen, Y. Song, D.M. McFarland, L.A. Bergman, A.F. Vakakis, Experimental study of non-linear effects in a typical shear lap joint
configuration, Journal of Sound and Vibration 277 (2004) 327–351.[18] J.P. Kahane, P.G. Lemariee-Rieusset, Seeries de Fourier et Ondelettes, Cassini, Paris, 1998 ISBN 284225001X.[19] B. Torreesani, Ondelettes, Analyse Temps-Freequence et Signaux Non-Stationnaires, Lecture Notes, 1997.[20] W.J. Staszewski, Identification of damping in MDOF systems using time-scale decomposition, Journal of Sound and Vibration 203 (2) (1997) 283–305.[21] J. Lardies, S. Gouttebroze, Identification of modal parameters using the wavelet transform, International Journal of Mechanical Sciences 44 (2002)
2263–2283.[22] T.-P. Le, P. Argoul, Continuous wavelet transform for modal identification using free decay response, Journal of Sound and Vibration 277 (2004)
73–100.[23] M. Ruzzene, A. Fasana, L. Garibaldi, B. Piombo, Natural frequencies and damping identification using wavelet transform: application to real data,
Mechanical Systems and Signal Processing 11 (2) (1997) 207–218.[24] P. Argoul, T.-P. Le, Instantaneous indicators of structural behaviour based on the continuous Cauchy wavelet analysis, Mechanical Systems and Signal
Processing 17 (1) (2003) 243–250.[25] R. Ghanem, F. Romeo, A wavelet-based approach for model and parameter identification of non-linear systems, International Journal of Non-Linear
